Final answer:
The steady-state level of income per capita in a Cobb-Douglas production function is found by setting the change in capital per capita to zero and solving the resultant equation.
Step-by-step explanation:
Finding the Steady-State Level of Income Per Capita
To find the steady-state level of income per capita with a Cobb-Douglas production function, we assume the production function has the form yₜ = Aₜkₜ^α, where yₜ is the income per capita, kₜ is the capital per capita, Aₜ is the level of technology, and α is the output elasticity of capital, which is between 0 and 1. In the steady state, the change in capital per capita Δkₜ is zero. From the given capital accumulation equation Δkₜ = syₜ−(n+δ)kₜ, setting Δkₜ equal to zero gives:
- 0 = sAₜkₜ^α - (n + δ)kₜ
- solve for kₜ by dividing both sides by s and factoring out kₜ:
- kₜ(sAₜkₜ^(α-1) - (n + δ)) = 0
- Since kₜ cannot be zero in this context (no capital would imply no output), we must have sAₜkₜ^(α-1) = n + δ.
- Rearrange to solve for kₜ:
- kₜ = (sAₜ / (n + δ))^(1 / (1−α))
- This represents the steady-state capital per capita.
- To find the steady-state income per capita, substitute the steady-state kₜ back into the production function:
- yₜ = Aₜ((sAₜ / (n + δ))^(1 / (1−α)))^α
- Simplify this expression to find the steady-state level of income per capita.
Understanding the steady-state income per capita is crucial as it represents a level where the economy can sustain itself without further accumulation or depletion of resources. It's an indicator of sustained economic growth and the well-being of a nation's population.